Engineering Optimization: Theory and Practice, Fourth Edition

34 Introduction to Optimization

whereEis the Young’s modulus andIis the area moment of inertia of the column

given by

I= 121 bd^3 (E 3 )

The natural frequency of the water tank can be maximized by minimizing−ω. With

the help of Eqs. (E 1 ) and (E 3 ), Eq. (E 2 ) can be rewritten as

ω=

[

Ex 1 x 23

4 l^3 (M+ 14033 ρlx 1 x 2 )

] 1 / 2

(E 4 )

The direct compressive stress(σc) n the column due to the weight of the water tanki

is given by

σc=

Mg

bd

=

Mg

x 1 x 2

(E 5 )

and the buckling stress for a fixed-free column(σb) s given by [1.121]i

σb=

(

π^2 EI

4 l^2

)

1

bd

=

π^2 Ex 22

48 l^2

(E 6 )

To avoid failure of the column, the direct stress has to be restricted to be less thanσmax

and the buckling stress has to be constrained to be greater tha n the direct compressive

stress induced.

Finally, the design variables have to be constrained to be positive. Thus the

multiobjective optimization problem can be stated as follows:

FindX=

{

x 1

x 2

}

whichminimizes

f 1 ( X)=ρlx 1 x 2 (E 7 )

f 2 ( X)=−

[

Ex 1 x 23

4 l^2 (M+ 14033 ρlx 1 x 2 )

] 1 / 2

(E 8 )

subjectto

g 1 (X)=

Mg

x 1 x 2

−σmax≤ 0 (E 9 )

g 2 (X)=

Mg

x 1 x 2

−

π^2 Ex 22

48 l^2

≤ 0 (E 10 )

g 3 ( X)=−x 1 ≤ 0 (E 11 )

g 4 ( X)=−x 2 ≤ 0 (E 12 )